Positron Emission Tomography (PET) is a nuclear medicine technique that is used to produce images for diagnosing the biochemistry or physiology of a specific organ, tumor, or other metabolically active site of the body.
FIG. 1 shows an example PET system. General information about PET imaging may be found in copending U.S. patent application Ser. No. 11/804,265 by Defrise et al. U.S. patent application Ser. No. 11/716,358 by Panin et al., both of which are hereby incorporated herein by reference in their entirety. A positron-emitting radioisotope 6 is introduced into the body 4 on a metabolically active molecule. When a positron encounters an electron, both are annihilated, yielding two gamma photons that travel in approximately opposite directions. A gantry 3 detects pairs of gamma rays 7 resulting from such annihilation events. The gantry 3 comprises a plurality of rings, with each ring comprising multiple scintillator crystals 8 and associated radiation detectors 5. Annihilation events are typically identified by a time coincidence between the detection of the two gamma photons by two oppositely disposed detectors, i.e., the gamma photon emissions are detected virtually simultaneously by each detector. When two oppositely traveling gamma photons strike corresponding oppositely disposed detectors to produce a time coincidence event, the photons identify a line of response (LOR) along which the annihilation event has occurred.
Images of metabolic activity in space are reconstructed by computer analysis. A processing unit (event detection unit) 1 determines and evaluates coincidence events generated by the pair of gamma rays and forwards this information to an image processing unit (computational unit) 2. Detector pairs associated to each line of response (LOR) produce many (e.g., millions of) coincidence events during a measurement.
FIG. 2 shows the conceptual basis for two-dimensional (2D) sinograms that are used for imaging. A sinogram 100 shows integrations along parallel LORs through all angles Φ, as shown on the left side of FIG. 2. The horizontal axis of the sinogram 100 represents distances on the projection axis s, and the vertical axis represents the different angles Φ of the projection of the scanned object f(x, y). The result of integrating along a particular LOR (i.e., for a particular value s and a particular angle Φ) is typically indicated by an intensity of a point in the s-Φ plane of the sinogram 100. For a given angle Φ, the result of integrating along various parallel LORs (i.e., varying s) yields a line in the s-Φ plane (with varying intensity at different points on the line), shown illustratively (not shown to scale in intensity) by a horizontal line in the sinogram 100. A single point 110 in the object 4 being defined by f(x, y) would thus be represented as a sinusoid in the sinogram 100 across varying angles Φ, as indicated by the dotted arrow.
FIG. 3 shows two different types of measurements into which coincidence events can be divided. The different types of measurements are based on the third dimension (z-axis) added by the plurality of rings in the gantry 3. In a 2D mode, the detectors 210 of the various rings are separated septa 220. Thus, in this mode, the scanner only collects coincidence events in direct planes 230 and cross planes 240, which are organized into direct planes. In a 3D mode, the scanner collects data from coincidence events that take place in all or most of the oblique planes 250. Each coincidence event depends on the particular location where the above described reaction took place and the pair of gamma photon was generated. Each gamma photon pair is therefore subject to location-dependent attenuation, resulting in projection data. PET data can be characterized to comprise direct (2D) and indirect (3D, also referred to as oblique) projection data.
In addition, modern PET systems may measure not only the attenuation but also the time of flight (TOF) of photons corresponding to a coincidence event, i.e., the time required for a photon to travel from its point of origin to the point of detection. TOF is dependent upon the speed of light c and the distance traveled. A time coincidence, or coincidence event, is identified if the time difference between the arrival of signals at a pair of oppositely disposed detectors is less than a coincidence time threshold τ.
FIG. 4 shows the conceptual basis of time of flight (TOF) determination. As illustrated in FIG. 4, if an annihilation event occurs at the midpoint of a LOR, the TOF of the gamma photon detected in detector A (TOF_A) is equal to the TOF of the gamma photon detected in detector B (TOF_B). If an annihilation event occurs at a distance Δx from the midpoint of the LOR, the difference between TOF_A and TOF_B is Δt=2Δx/c, where c is the speed of light. If d is the distance between the detectors, the TOF difference Δt could take any value from −d/c to +d/c, depending on the location of the annihilation event.
According to the principles of TOF positron emission tomography (TOF-PET), measurement of the difference Δt between the detection times of the two gamma photons arising from the positron annihilation event allows the annihilation event to be localized along the LOR with a resolution of about 75-120 mm corresponding to full width at half maximum (FWHM), assuming a time resolution of 500-800 picoseconds (ps). Though less accurate than the spatial resolution of the scanner, this approximate localization is effective in reducing the random coincidence rate and in improving both the stability of the reconstruction and the signal-to-noise ratio (SNR), especially when imaging large objects.
TOF acquisition results in additional data dimensions that significantly increase data size. One practical existing approach is called list mode reconstruction (see D. L. Snyder and D. G. Politte, “Image reconstruction from list-mode data in an emission tomography system having time-of-flight measurements,” IEEE Trans. Nucl. Sci., vol. 30, pp. 1843-1849, 1983”). However, list mode reconstruction depends on the number of registered events and therefore is time consuming for high count studies. In addition, list-mode reconstruction is only possible with certain algorithms. An alternative solution exploits the redundancy of TOF information to compress data without loss of resolution (see S. Vandenberghe, M. E. Daube-Witherspoon, R. M. Lewitt and J. S. Karp, “Fast reconstruction of 3D TOF PET data by axial rebinning and transverse mashing,” Phys. Med. Biol., vol. 51, 1603-1621, 2006). Such compression may consist of axial rebinning and azimuthal mashing, resulting in histogrammed data that can be reconstructed using any algorithm and in a manner that is independent of the acquisition time.
Prior art methods based on Fourier transforms use redundancy to rebin a 3D data set into 2D data, similarly to what is done for non-TOF PET data (see M. Defrise, C. Michel, M. Casey and M. Conti, “Fourier rebinning of time-of-flight PET data,” Phys. Med. Biol., vol. 50, pp. 2749-2763, 2005 and S. Cho, S. Ahn, Q. Li and R. Leahy, “Exact and approximate Fourier rebinning of PET data from time-of-flight to non time-of-flight”, Phys. Med. Biol., vol. 54, pp. 467-484, 2009). A disadvantage of the Fourier methods is the necessity to synthesize missing data due for example to gaps between blocks. In addition, these methods assume a parallel beam data organization, which is not natural for a cylindrical scanner. Pure axial rebinning methods exist for TOF data, such as Single Slice Rebinning (TOF-SSRB) (see N. A. Mullani, W. H. Wong, R. Hartz, K. Yerian, A. Philippe and K. L. Gould, “Sensitivity improvement of TOF PET by the utilization of the inter-slice coincidences,” IEEE Trans. Nucl. Sci., vol. 29, pp. 479-83, 1982). Such rebinning is performed independently for each azimuthal and radial coordinates and is suitable for LOR data. Recently, an exact axial rebinning for TOF data was derived, based on Consistency Conditions (CCs), which generalize John's equation. Unfortunately, these CCs contain second order derivatives of the measured data, resulting in noise amplification during rebinning. This fact led to the investigation of a generalized rebinning with a noise-resolution tradeoff property (see M. Defrise, V. Panin, C. Michel, M. Casey, “Continuous and Discrete Data Rebinning in Time-of-Flight PET”, IEEE Trans. Med. Imag., vol. 27, pp. 1310-1322, 2008). 3D reconstruction is more expensive in terms of time and resources than the rebinned based methods but advantageously allows accurate statistical modeling.
The Fourier transform based inverse rebinning methods in (X. Liu, M. Defrise, P. E. Kinahan, C. Michel, M. Sibomana, and D. Townsend, “Exact rebinning methods for 3D PET,” IEEE Trans. Med. Img., vol. 18, pp. 657-664, 1999) and (S. Cho, Q. Li, S. Ahn, and R. M Leahy, “Iterative Image Reconstruction Using Inverse Fourier Rebinning for Fully 3-D PET,” IEEE Trans. Med. Imag., vol. 26, pp. 745-756, 2007) allow fast forward projection of non-TOF data in a parallel beam geometry. In addition, these methods permit point spread function (PSF) modeling despite being based on the line integral model that is implied by the Fourier relations (see A. Reader, P. J. Julyan, H. Williams, D. L. Hastings, and J. Zweit, “EM Algorithm System Modeling by Image-Space Techniques for PET Reconstruction,” IEEE Trans. Nucl. Sci., vol. 50, pp 1392-1397, 2003 and A. Alessio, P. Kinahan, T. Lewellen, “Modeling and incorporation of system response functions in 3D whole body PET”, IEEE Trans. Med. Imag., vol. 25, pp. 828-837, 2006). The accuracy of the Fourier methods, however, depends on having a fine sampling of the 3D data. The method in the Liu et al. reference, in particular, disadvantageously requires a fine azimuthal sampling, which is incompatible with the ordered subsets (OS) iterative methods, where each data subset is significantly undersampled.